Quick and Dirty Introduction to
Mott Insulators
Introduction to Solid State Physics
http://www.physics.udel.edu/~bnikolic/teaching/phys624/phys624.html
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Weakly correlated electron liquid:
Coulomb interaction effects
When local perturbation  U (r ) potential is
switched on, some electrons will leave this
region in order to ensure constant  F 
(chemical potential is a thermodynamic
potential; therefore, in equilibrium it must be
homogeneous throughout the crystal).
 n(r)  eD( F ) U (r)
assume: e U (r)
F
f ( , T  0)   ( F   )
PHYS 624: Quick and dirty introduction to Mott Insulators
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Thomas –Fermi screening
•Except in the immediate vicinity of the
perturbation charge, assume that  U (r) is caused
by the induced space charge → Poisson equation:
 r / rTF
1



e
2  2
r 2   U (r ) 
r r r
r
rTF 
rTF
0
e2 D( F )
in vacuum: D( F )  0,  U (r) 
e n(r )
  U (r )  
0
2
1 n 
 
2  a03 
1/ 6
4 2 0
, a0 
me2
nCu  8.5 1023 cm3 , rTFCu  0.55Å
q
4 0

2
1/3
2/3
1/3
3 n
1 2m 2 2/3
4
n
D( F ) 
 2 2  3 n  ,  F 
3 2 n   rTF2   3 2 

2  F 2
2m

a0
PHYS 624: Quick and dirty introduction to Mott Insulators
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Mott Metal-Insulator transition
2
TF
r
n 1/ 3
1 a0
1/ 3
4n
4a0
a02
•Below critical electron concentration, the potential well of the
screened field extends far enough for a bound state to be
formed → screening length increases so that free electrons
become localized → Mott Insulators (e.g., transition metal
oxides, glasses, amorphous semiconductors)!
PHYS 624: Quick and dirty introduction to Mott Insulators
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Metal vs. Insulator
Fundamental requirements for
electron transport in Fermi
systems:

•
quantum-mechanical states
for electron-hole
excitations must be
available at energies
immediately above the
ground state since the
external field provides
vanishingly small energy
•
these excitations must
describe delocalized
charges that can contribute
to transport over the
macroscopic sample sizes.

T
PHYS 624: Quick and dirty introduction to Mott Insulators
T
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Metal-Insulator Transitions
Mott Insulator: A solid in which
strong repulsion between the particles
impedes their flow → simplest cartoon
is a system with a classical ground
state in which there is one particle on
each site of a crystalline lattice and
such a large repulsion between two
particles on the same site that
fluctuations involving the motion of a
particle from one site to the next are
suppressed.
From weakly correlated Fermi liquid to strongly correlated Mott insulators
INSULATOR
nc
STRANGE METAL
STRONG CORRELATION
PHYS 624: Quick and dirty introduction to Mott Insulators
2nc
F. L. METAL
n
WEAK CORRELATION
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Energy band theory
Electron in a periodic potential (crystal)
 energy band (  (k )  2t cos(ka) : 1-D tight-binding band)
N=1
N=2
N=4
N=8
N = 16
N=
EF
kinetic energy gain
PHYS 624: Quick and dirty introduction to Mott Insulators
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Band (Bloch-Wilson) insulator
Wilson’s rule 1931: partially filled energy band  metal
otherwise  insulator
metal
insulator
semimetal
Counter example: transition-metal oxides, halides, chalcogenides
Fe: metal with 3d6(4sp)2
FeO: insulator with 3d6
PHYS 624: Quick and dirty introduction to Mott Insulators
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Mott gedanken experiment (1949)
electron transfer integral t
energy cost U
d
atomic distance
d   (atomic limit: no kinetic energy gain): insulator
d  0 : possible metal as seen in alkali metals
Competition between W(=2zt) and U
 Metal-Insulator Transition
e.g.: V2O3, Ni(S,Se)2
PHYS 624: Quick and dirty introduction to Mott Insulators
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Mott vs. Bloch-Wilson insulators
•Band insulator, including familiar semiconductors, is state produced by a
subtle quantum interference effects which arise from the fact that electrons are
fermions.
•Nevertheless one generally accounts band insulators to be “simple” because
the band theory of solids successfully accounts for their properties
•Generally speaking, states with charge gaps (including both Mott and BlochWilson insulators) occur in crystalline systems at isolated “occupation
numbers”    * where  * is the number of particles per unit cell.
•Although the physical origin of a Mott insulator is understandable to any child,
other properties, especially the response to doping    *   are only
partially understood.
•Mott state, in addition to being insulating, can be characterized by: presence or absence of
spontaneously broken symmetry (e.g., spin antigerromagnetism); gapped or gapless low energy
neutral particle excitations; presence or absence of topological order and charge fractionalization.
PHYS 624: Quick and dirty introduction to Mott Insulators
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Theoretical modeling: Hubbard Hamiltonian
Hubbard Hamiltonian 1960s:
on-site Coulomb interaction is most dominant
band structure
correlation
e.g.: U ~ 5 eV, W ~ 3 eV for most 3d transition-metal oxide such as
MnO, FeO, CoO, NiO : Mott insulator
 Hubbard’s solution by the Green’s function
decoupling method
 insulator for all finite U value
 Lieb and Wu’s exact solution for the ground
state of the 1-D Hubbard model (PRL 68)
 insulator for all finite U value
PHYS 624: Quick and dirty introduction to Mott Insulators
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Trend in the Periodic Table
U
U
PHYS 624: Quick and dirty introduction to Mott Insulators
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Solving Hubbard model in
 dimensions
•In -D, spatial fluctuation can be neglected.
→ mean-field solution becomes exact.
•Hubbard model → single-impurity Anderson
model in a mean-field bath.
•Solve exactly in the time domain
→ “dynamical” mean-field theory
PHYS 624: Quick and dirty introduction to Mott Insulators
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From non-Fermi liquid metal to Mott
insulator
NOTE: DOS defined
even though there are no
fermionic quasiparticles.
Model: Mobile spin-up
electrons interact with
frozen spin-down
electrons.
PHYS 624: Quick and dirty introduction to Mott Insulators
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Experiment: Photoemission Spectroscopy
h (K,) > W
Einstein’s photoelectric effect
e- (Ek,k,)
N-particle
(N1)-particle
Sudden approximation
EfN 1
E iN
P(| i   | f )
Photoemission current is given by:
1
2
 EiN / k BT
A ( )   e
 f | Tr | i   (  E fN 1  EiN )
Z i, f
PHYS 624: Quick and dirty introduction to Mott Insulators
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Mott Insulating Material: V2O3
surface-layer thickness =
2.44Å
–
(1012) cleavage plane
side view

c = 14.0 Å

a = 4.95 Å
Vanadium
Oxygen
PHYS 624: Quick and dirty introduction to Mott Insulators
top view
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Phase diagram of V2O3
PHYS 624: Quick and dirty introduction to Mott Insulators
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Bosonic Mott insulator in optical
lattices
•Superfluid state with
coherence, Mott Insulator
without coherence, and
superfluid state after
restoring the coherence.
PHYS 624: Quick and dirty introduction to Mott Insulators
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